Using your Head is Permitted

February 2014 riddle

UPDATE (6 February): Some solvers have asked me if I consider
searching in Wikipedia (or elsewhere) for how to solve the inverse square
equation in preparation for solving the inverse cubic equation to be in
violation of what I consider an "original" solution. The answer is that
originality of a solution is as defined here,
a link that I point to every month. I believe it covers the current situation
quite explicitly, and the answer is: yes, it is in violation. If you looked up
how to solve for the inverse square case after reading this riddle, you are
most welcomed to enjoy the riddle, but don't ask to be listed alongside people
who have solved it entirely by themselves.

One fine evening, on January of 1684, three men came out of a meeting of The
Royal Society and headed out to a nearby coffee shop. Based on the conversation
that ensued and the outrageous wager the three later made over a sum of money,
one can only suspect that what they drank was somewhat stronger than coffee.

The three were Edmond Halley, Christopher Wren and Robert Hooke, and the topic
under discussion was the hot one of the day: finding a formulation of gravity
that would explain Johannes Kepler's laws of motion
(which were published earlier in the same century).

Hooke bet that he could do it. Wren put up cash money to say that he can't.

Ultimately, Hooke never was able to collect on that bet, but Halley had the
good sense
to turn this problem to someone else, Isaac Newton, who managed to provide the
solution: if a planet orbits a stationary star, and if the force of gravity
applied by the star on the planet is directed towards the star and its
magnitude is inversely proportional to the square of the distance between the
planet and the star, the planet will follow Kepler's laws of motion. Namely,
it will orbit in an elliptical path. (In the process, Newton also invented
both Calculus and science as we know it. However, history books are silent on
the question of whether he made Wren cough up any cash.)

Newton's discovery is nowadays referred to as "the inverse square law of
gravity".

This month's question: suppose gravity would not have worked according to an
inverse square law. Suppose, instead, that gravity would have worked according
to an inverse cubic law. What stable orbits would planets around
stationary stars hold, in such a universe?

To be clear, "stable orbit" means here a locus in space that describes a
closed path around the star (not hitting into the star) that the planet will
follow. There is no requirement for this orbit to be "stable" in the sense of
being a "stable equilibrium".

Solvers can get an asterisk next to their name if they can, additionally,
provide substantial additional characterization of the motions of such planets.